Quantum Information for Quantum Cats (24)

Quantum Phase Oracle

We now want to use this quantum oracle U_f fro function f(x). It is however more convenient to define another unitary operator V_f by 　

with previously defined W={{1,0,0,0}{0,-1,0,0},{0,0,01,0},{0,0,0,-1}} which corresponds to multiplying the sign (-1) when the lowest qubit has value 1. Consider the operation of this derivative operator on thhe state | x, 0 >, which is given by

The result is that we have the same state | x, 0 > with a factor (-1)^f(x) multiplied. We therfore call this V_f a phase oracle. of funtion f. Again, explicit construction for single qubit x case is helpful

Since the oracle qubit is unaltered to be | 0 > throughout the operation of V_f, we might just consider V_f to be operating on input state | x > only, and write it as

For the case of single qubit x, above expressions of V_f are simplified into

whose two qubit generalization, readers are again encouraged to work out.